【微分積分】3-4-2 ライプニッツの公式｜問題集
1.次の関数の\(n\)階導関数を答えなさい。
(1)\(f(x)=x^2e^{2x}\)
\(\displaystyle f^{(n)}(x)=\sum_{k=0}^{n}{}_{n}\mathrm{C}_k(x^2)^{(k)}(e^{2x})^{(n-k)}\)
\(=x^2(e^{2x})^{(n)}+{}_{n}\mathrm{C}_1(x^2)'(e^{2x})^{(n-1)}\)
\(\ \ \ \ \ +{}_{n}\mathrm{C}_2(x^2)''(e^{2x})^{(n-2)}\)
\(\displaystyle =x^2・2^ne^{2x}+n・2x・2^{n-1}e^{2x}\)
\(\displaystyle \ \ \ \ \ +\frac{n(n-1)}{2}・2・2^{n-2}e^{2x}\)
\(=e^{2x}2^{n-2}\{4x^2+4nx+n(n-1)\}\)
\(=x^2(e^{2x})^{(n)}+{}_{n}\mathrm{C}_1(x^2)'(e^{2x})^{(n-1)}\)
\(\ \ \ \ \ +{}_{n}\mathrm{C}_2(x^2)''(e^{2x})^{(n-2)}\)
\(\displaystyle =x^2・2^ne^{2x}+n・2x・2^{n-1}e^{2x}\)
\(\displaystyle \ \ \ \ \ +\frac{n(n-1)}{2}・2・2^{n-2}e^{2x}\)
\(=e^{2x}2^{n-2}\{4x^2+4nx+n(n-1)\}\)
(2)\(f(x)=x^3e^x\)
\(\displaystyle f^{(n)}(x)=\sum_{k=0}^{n}{}_{n}\mathrm{C}_k(x^3)^{(k)}(e^x)^{(n-k)}\)
\(=x^3(e^x)^{(n)}+{}_{n}\mathrm{C}_1(x^3)'(e^{x})^{(n-1)}\)
\(\ \ \ \ \ +{}_{n}\mathrm{C}_2(x^3)''(e^{x})^{(n-2)}+{}_{n}\mathrm{C}_3(x^3)'''(e^{x})^{(n-3)}\)
\(=x^3・e^x+n・3x^2・e^x\)
\(\displaystyle \ \ \ \ \ +\frac{n(n-1)}{2}・6x・e^x+\frac{n(n-1)(n-2)}{6}・6・e^x\)
\(=e^x\{x^3+3nx^2+3n(n-1)x+n(n-1)(n-2)\}\)
\(=x^3(e^x)^{(n)}+{}_{n}\mathrm{C}_1(x^3)'(e^{x})^{(n-1)}\)
\(\ \ \ \ \ +{}_{n}\mathrm{C}_2(x^3)''(e^{x})^{(n-2)}+{}_{n}\mathrm{C}_3(x^3)'''(e^{x})^{(n-3)}\)
\(=x^3・e^x+n・3x^2・e^x\)
\(\displaystyle \ \ \ \ \ +\frac{n(n-1)}{2}・6x・e^x+\frac{n(n-1)(n-2)}{6}・6・e^x\)
\(=e^x\{x^3+3nx^2+3n(n-1)x+n(n-1)(n-2)\}\)
(3)\(f(x)=x^2\cos2x\)
\(\displaystyle f^{(n)}(x)=\sum_{k=0}^{n}{}_{n}\mathrm{C}_k(x^2)^{(k)}(\cos2x)^{(n-k)}\)
\(\displaystyle =x^2(\cos2x)^{(n)}+{}_{n}\mathrm{C}_1(x^2)'(\cos2x)^{(n-1)}\)
\(\ \ \ \ \ +{}_{n}\mathrm{C}_2(x^2)''(\cos2x)^{(n-2)}\)
\(\displaystyle =x^2・2^n\cos\left(2x+\frac{n\pi}{2}\right)\)
\(\displaystyle \ \ \ \ \ +n・2x・2^{n-1}\cos\left(2x+\frac{n-1}{2}\pi\right)\)
\(\displaystyle \ \ \ \ \ +\frac{n(n-1)}{2}・2・2^{n-2}\cos\left(2x+\frac{n-2}{2}\pi\right)\)
\(\displaystyle =2x^2・2^{n-1}\cos\left(2x+\frac{n\pi}{2}\right)\)
\(\displaystyle \ \ \ \ \ +2nx・2^{n-1}\sin\left(2x+\frac{n\pi}{2}\right)\)
\(\displaystyle \ \ \ \ \ -\frac{n(n-1)}{2}・2^{n-1}\cos\left(2x+\frac{n\pi}{2}\right)\)
\(\displaystyle =2^{n-2}\{4x^2-n(n-1)\}\cos\left(2x+\frac{n\pi}{2}\right)\)
\(\displaystyle \ \ \ \ \ +2^nnx\sin\left(2x+\frac{n\pi}{2}\right)\)
\(\displaystyle =x^2(\cos2x)^{(n)}+{}_{n}\mathrm{C}_1(x^2)'(\cos2x)^{(n-1)}\)
\(\ \ \ \ \ +{}_{n}\mathrm{C}_2(x^2)''(\cos2x)^{(n-2)}\)
\(\displaystyle =x^2・2^n\cos\left(2x+\frac{n\pi}{2}\right)\)
\(\displaystyle \ \ \ \ \ +n・2x・2^{n-1}\cos\left(2x+\frac{n-1}{2}\pi\right)\)
\(\displaystyle \ \ \ \ \ +\frac{n(n-1)}{2}・2・2^{n-2}\cos\left(2x+\frac{n-2}{2}\pi\right)\)
\(\displaystyle =2x^2・2^{n-1}\cos\left(2x+\frac{n\pi}{2}\right)\)
\(\displaystyle \ \ \ \ \ +2nx・2^{n-1}\sin\left(2x+\frac{n\pi}{2}\right)\)
\(\displaystyle \ \ \ \ \ -\frac{n(n-1)}{2}・2^{n-1}\cos\left(2x+\frac{n\pi}{2}\right)\)
\(\displaystyle =2^{n-2}\{4x^2-n(n-1)\}\cos\left(2x+\frac{n\pi}{2}\right)\)
\(\displaystyle \ \ \ \ \ +2^nnx\sin\left(2x+\frac{n\pi}{2}\right)\)
次の学習に進もう！